On manifolds of tensors of fixed TT-rank

By Sebastian Holtz, Thorsten Rohwedder, Reinhold Schneider. 2012

286 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Nuclear and High Energy Physics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Initially proposed for low-rank matrix manifolds in [167], dynamical low-rank methods have been extended to low-rank tensors in Tucker [168, 200], TT/MPS [119, 124, 164, 187], and HT [9, 187, 257] decomposition.”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, methodology): “6) inductively, one can characterize the TT ranks as follows [21].”
Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors (2013, methodology): “Analogous to the Tucker format, both HT and TT allow the definition of a rank vector k (to be specified in section 2), and the manifold of tensors of fixed HT or TT rank can be shown to be an embedded manifold [12, 30], to which the above dynamical tensor approximation approach may then be applied.”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, background): “The only problem is that the longer the time interval one considers, the more significantly different snapshots have to be stored in one QTT-tensor, resulting in the larger QTTranks to keep for the same accuracy.”
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation (2011, background): “Sketch of numerical TT/QTT approximation The manifold [14] of rank-r TT tensors in W n is known to be closed in the Frobenius norm [24].”
On the approximation of high-dimensional differential equations in the hierarchical Tucker format (2013, unknown): “, in [3].”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, unknown): “(100) For each i, Ci−1ii ∗ is the space of all elements with full multilinear rank: Cri−1×qi×ri ∗ := {Ai ∈ Cri−1×qi×ri : rank([Ai] mi−1 ) = ri−1, rank([Ai] mi−1,αi) = ri} (101) Let it be noted that this space is a smooth manifold(7).”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “Using these projectors the orthogonal projector onto the tangent space, PTXMr(Z), can be written as [Holtz et al., 2012b, Lubich et al., 2015] PTXMr(Z) = N ÿ n=1 PďnX (P ěn+1 X (Z)) , (3.110) where PďnX (Z) = Ten(PďnZăną) , (3.111) Pěn+1X (Z) = Ten(ZănąPěn+1), (3.112) and Ten(Z) is an inverse…”
Fast and accurate 3D tensor calculation of the Fock operator in a general basis (2012, background): “…= J(C) +K(C), (2.9) yielding the Galerkin equation for the Hartree-Fock eigenvalue problem, FC = SCΛ, Λ = diag(λ1, ..., λNb), (2.10) C∗SC = INb , INb ∈ RNb×Nb where the second equation represents the orthogonality constraints ∫ R3 ψiψj = δij and S = { ∫ R3 gν(x)gµ(x)dx} is the Galerkin mass matrix.”